R e a c t o r s & its kinetics

1. R E A C T O R S
&
REACTOR KINETICS
BY
R A CHRISTIAN, Ph. D.
Assistant Professor, Civil Engineering Department
S V National Institute of Technology, Surat
5th October 2009

2. CONCEPT OF REACTORS
Reactors (Treatment Units)
The units or vessels that hold wastewater for treatment by chemical or biological processes are
normally called as reactors and the units that are used for separation of solids from liquid by
settling or flotation are termed as basins or tanks. However, in practice the terms basins,
tanks, vessels or reactors are used interchangeably.
The reactors may be of any shape but mostly rectangular or circular reactors are used in
wastewater treatment. The size (capacity or volume) of a reactor, more particularly in biological
processes, normally depends on the treatment system selected, order of reaction rate assumed
and the flow conditions (hydraulic regime) that will prevail in the reactor.
Types of Reactors
Depending upon the flow and operating conditions and the method of mixing of the wastewater
therein, the reactors have been classified as under :
Continuous - Flow Stirred Tank Reactor (CFSTR)
Plug - Flow Reactor (PFR)
Completely Mixed Batch Reactor (CMBR)
Arbitrary - Flow Reactor (AFR)
Fluidized Bed Reactor (FBR)
Packed Bed Reactor (PBR)
Sequencing Batch Reactor (SBR)
As the selection of a reactor and its design for achieving the desired degree of treatment
requires a clear understanding of each of the above classified reactors, they have been briefly
described in this presentation.

3. Continuous - Flow Stirred Tank Reactor (CFSTR)
CFSTR is also called Completely Mixed Reactor. As the flow of wastewater is continuous in such type of
reactors, the reactants entering the reactor and the products flowing out from the reactor is considered
as continuous. It is also assumed that the contents are distributed throughout the tank as soon as the
flow enters the reactor and their uniform concentrations are maintained in the reactor operating under
steady state conditions. Fig. 1 given below shows the schematic of CFSTR.
Notations in figure represent,
V Ж Ce
Q
Co
V = Reactor volume
Q0 = Wastewater flow rate into and out of reactor
Co = Initial reactant concentration in influent
Ce = Final Reactant concentration in effluent
= Reactant concentration in reactor
Q
Ce
Fig. 1 Schematic of CFSTR

4. The equations for HRT and effluent reactant concentrations are derived from the mass balance of
reactant as given below:
Net rate of change
in mass of reactant
within the reactor
=
Rate of increase
in mass of
reactant
due to its
presence in the
influent
+
Rate of decrease
in mass of reactant
due to its removal in the
effluent
-
Rate of
decrease
in mass of
reactant
due to
reaction of
reactants in
the reactor
Or mathematically,
V é dc ù êë úû =QC -Q C - V é dc
ù
dt o e e
êë dt
úû net r
For first order reaction kinetics,
V dc =QC -Q C - VKC
é ù
êë úû o e e e
net
dt

5. Or mathematically,
ù
úû
1 + K é
V
êë
=
Q
1
C
C
o
or C = 1
C 1+(K×t )
e e
o CFSTR
or C = 1
C 1+(K×V / Q)
e
o CFSTR
é ù
ê ú
ë û
or V = Q C 0
- 1
CFSTR
e
K C
where : K = reaction rate constant, and
t = reaction time (hydraulic retention time) to achieve desired reactant concentration
V = volume of reactor
Q = flow of wastewater
Similarly, the equation derived to obtain the effluent concentration for second order reaction kinetics is
given by
C =
1
C 1 (K t C )
e
+ ´ ´
o CFSTR e
or C = 1
C 1+(K×(V /Q)×C )
e
o CFSTR e
é ù
ê ú
ë û
or V = Q C -C
0 e
CFSTR 2
e
K C

6. the hydraulic retention time for CFSTR is given by
é ù
ê ú
ë û
t = 1 C o
-1
CFSTR
e
K C
for first order reaction
é ù
ê ú
ë û
and t = 1 C -C
0 e
CFSTR 2
e
K C
for second order reaction
é ù
ê ú
ë û
t = 1 C 0
-1
CFSTR
K C C
e e
where, K = second order rate constant, [(mg/L)xd]-1
Illustrative Example
A wastewater is being treated in a CFSTR following first order reaction
kinetics with a reaction rate constant equal to 0.15 day-1. For a reactor
volume of 50 m3, what should be the flow rate to achieve 96% treatment
efficiency? For this flow rate, compute the reactor volume if the desired
treatment efficiency is 98%?
Flow rate of wastewater required to achieve 96% efficiency in CFSTR of 50 m3 capacity is 0.313
m3/day. Now, for same operating conditions, when the desired treatment efficiency of 98% is to
be achieved, the volume required will be 102.25 m3 .
Thus, the capacity of the reactor will be almost doubled (from 50 m3 to 102 m3) when
treatment efficiency is increased from 96% to 98% for the given conditions of wastewater
treatment.

7. NOTE: It is not economical to increase the volume of reactor by two times just to achieve 2% more
treatment efficiency.
Plug - Flow Reactor (PFR)
In a plug flow reactor, the content of wastewater follows the principle of 'first - in - first - out'. So,
the particles pass through the tank in the same order or sequence in which they enter the tank
and longitudinal mixing is assumed to be almost negligible.
The concentration of a reactant varies with time and along the length of the reactor. Fig. 2 given
below shows the schematic of a plug flow reactor.
V
Fig.2 Schematic for PFR
Q
C0
Q
Ce
The equations for HRT and concentration of reactant in effluent are derived from mass-balance
of a reactant at steady state conditions as under:
Change in the
concentration of reactant
due to reaction of reactant
in time, dt
=
Change in the
concentration of reactant
due to change in position
of fluid element in time, dt

8. i.e. - dc = dx (-ve sign implies a decrease in reactant concentration)
dt v
where, v = velocity of flow through reactor
dx = differential change in distance along the length of reactor
Integrating the left hand side of the equation between concentration limits C0 to Ce and
integrating right hand side of the equation for lengths zero to L and substituting the value of v/Q
for L/v; we get the equations to determine HRT and volume of reactor as given below:
For first order reaction kinetics,
é ù
ê o
ú
ë û
PFR
e
t = 1 lnC
K C
é ù
ê ú
ë û
V = Q lnC
o
e
K C
Similarly for second order reaction kinetics,
é ù
ê ú
ë û PFR
t = 1 1 - 1 for second order reaction
K C C
e 0
é ù
ê ú
ë û
t = 1 C 0
-1
PFR
K C C
0 e
where, K = second order rate constant, [(mg/L)xd]-1
é ù
ê ú
ë e 0 û
and V = Q 1 - 1 for second order reaction
K C C

9. Illustrative Example
If the rate of reaction in the system is of second order, compare
the required volume of continuous flow stirred tank reactor and
the volume of plug flow reactor to achieve 94 % reduction in the
reactant concentration in the system.
VCFSTR = 16 x VPFR
NOTE: In general, the volume required for CFSTR will always be more than that of PFR.
Completely Mixed Batch Reactor (CMBR)
Completely Mixed Batch Reactor is a closed system where no flow is added or allowed to go out
during designed reaction time (detention period). The reactants are added to the reactor when it
is empty and the contents are withdrawn after the reaction period is over. In CMBR, it is assumed
that the reaction kinetics are of first order and at a given instant of time, the reactant
concentration is uniform throughout the reactor. Fig. 3 shows the schematic of CMBR.
V, Ж C1
Q0
C0
Q0
C1
Fig. 3 Schematic of CMBR

10. The mass-balance for a reactant in CMBR can be expressed as :
Rate of change in the mass
of reactant within the
reactor
=
Rate of the reaction of
reactant within the reactor
For first - order reaction kinetics, mathematically,
V dC V dC V(KC) (4.31)
´ æ ö ö çè dt ÷ø = ´æ çè = net dt
÷ø reaction
we get the following equation to determine HRT and volume of the reactor,
ù
úû
é
lnC
K
êë
t =
1
o
e
CMBR C
Where, Co = initial reactant concentration
Ce = desired or final reactant concentration
K = reaction rate constant
é æ öù
ê ç e
÷ú
ë è 0
øû
Q ln C
and, V = K C (as t = V/Q)

11. NOTE: Application of CMBR is limited in biological wastewater treatment. However, its use
is more in bench scale laboratory studies and in digestion of sludge.
Arbitrary Flow Reactor (AFR)
A PFR designed with dispersion of flow is called an Arbitrary Flow Reactor. In
practice, some intermediate amount of intermixing will always occur. The equation
developed by Whener and Wilhem for such intermediate mixing occurring in AFR is
as given below:
1
Co 4ae2d
= C a -
a e (1 + a)2e2d - (1 -
a)2e2d
where, a 1 4 Ktd and d D Dt
= + = = 2
vL L
here, K = reaction rate constant, (time-1)
t = hydraulic retention time
d = dispersion number or diffusivity constant (dimensionless)
= 0 for PFR
= α (infinity) for completely mixed system
D = axial dispersion coefficient (area/time)
v = fluid velocity (length/time)
L = characteristic length of travel path of particle

12. Reactors in Series :
In the design of wastewater flow treatment system, sometimes, either the same or
combination of different types of reactors are required to be used in series. The
reactors provided in series may or may not be of equal size and may be operating on
different types of processes. Fig. 4 given below shows two continuous flow stirred
tank reactors in series.
¥
V, C1
Q0
C0
Q0
C1
¥
V, C2
Q0
C2
Fig. 4 Schematic of two CFSTRs in series

13. Assuming first - order reaction kinetics and 'n' number of equal sized CFSTR, we get
the equation for detention time and thereby the total volume of reactors as follows:
ù
ú ú ú
CFSTRinseries detention time in one reactor
û
é
æ
ê ê ê
ë
ö
1
- ÷ ÷ø
ç çè
C
t 1
= 1
C
K
n
o
e
ù
ú ú ú
CFSTR to determine detention time in ‘n’ reactors
û
é
æ
ê ê ê
ë
ö
1
- ÷ ÷ø
ç çè
C
n t n
´ = 1
C
K
n
o
e
ù
ú ú ú
o to determine volume of ‘n’ reactors
û
é
æ
ê ê ê
ë
ö
1
- ÷ ÷ø
ç çè
C
n
or n V
´ = 1
C
K
Q
n
e
Illustrative Example
Calculate and compare the volume of the reactor(s) required to achieve 90%
reduction of a reactant in a flow of 1000 m3/d for the following conditions:
i) Single CFSTR is used
ii) Four CFSTRs are used in series, and
iii) Single PFR is used.
Assume the reaction rate constant, K = 0.5 day--1.

14. Assuming first order reaction kinetics for all three given conditions,
when a single CFSTR is used, VCFSTR = 18000.0 m3
when four CFSTR are used in series, Total volume = 6224.0 m3 (say)
when a single PFR is used, V = 4600.0 m3
COMMENT: Of the above 3 conditions, least volume of reactor is required for single
PFR and less total volume is required when four CFSTRs are provided in series
instead of one CFSTR.
More the number of CFSTR in series, smaller will be the total reactor volume for the system and
the system approaches PFR with increase in number of reactors in series.

15. Illustrative Example
A reactor system reduces the influent reactant concentration from
200 mg/L to 20 mg/L with a detention time of 20 days. Assuming that
the reaction rate is of first order, determine the value of K for a)
CFSTR and b) PFR. Give your comments on the results.
For CFSTR, KCFSTR = 0.45 day –1
For PFR, K = 0.115 day –1
K 0.45 3.91 4.0
K 0.115
CFSTR = = =
PFR
COMMENTS: The values of reaction rate constants show that the reduction in
reactant concentrations is about 4 times faster in CFSTR than in PFR, when
similar conditions of flow, HRT and effluent concentration of reactant are
maintained in both the reactor systems.

16. Fluidized Bed Reactors (FBR)
A reactor in which the filled packing material expands and gets fluidized when the
wastewater to be treated moves upward in the reactor is called a FBR. Normally,
air is also introduced along with the influent flow from the inlet. Fig.5 shows the
schematic of FBR
Effluent
Fluidized bed of
packing material
Influent
Gas(es)
Fig. 5 Schematic of a FBR
Such reactors are becoming popular to treat wastewaters biologically either in
aerobic or anaerobic conditions. They are also used for sludge treatment and
removal of dissolved gases.

17. Packed Bed Reactors (PBR)
A reactor in which the filled inert packing material for the growth of biomass is
kept packed (or fixed) is called a PBR. The flow of wastewater through the reactor
may be upward or downward as shown in Fig. 6. The packing material commonly
used is slag, rock or ceramic. However, the use of plastic as packing material,
with various configuration and large specific area, is now more common.
a) Downflow PBR
Bed of
packing
material
Effluent
Influent
Gas(es) Gas(es)
Effluent
Influent
b) Upflow PBR
Fig. 6 Schematic for PBR
NOTE: When a reactor is completely filled with packing media with respect to flow, it is
known as Anaerobic Reactor (or Filter).

18. Sequencing Batch Reactors (SBR)
This is a fill and draw type of reactor working on the principle of an activated
sludge process where reactions for aeration and waste conversion and
clarification of effluent occur in the same reactor but in sequencing steps.
Operational steps:
The reactor is first filled with the wastewater up to the desired volume and the flow
is stopped.
The content of wastewater is then aerated and mixed for the designed time
period.
Aeration is then stopped and clarification or sedimentation of biomass is carried
out to separate the sludge.
The clarified effluent is then withdrawn (or decanted) from the reactor.
Finally the deposited sludge is removed from the bottom of reactor.

19. Fig. 7 shows the operating steps of SBR used for activated sludge process
system.
Aeration of W/W
Influent
Q, S0
Step – 1
Filling up the reactor
Step – 2
Reaction takes place for time
t
Step – 3
Settling of sludge (clarification)
Step – 4
Effluent
Q, Se
Removal of clarified effluent
Step – 5
Qw
Removal of sludge
Fig. 7 Operating steps of SBR
SBR system does not require recycling of the activated sludge to maintain MLSS in the
reactor. The sludge wasting depends on performance requirements.
TIP : To operate the SBR system on a continuous basis, two or more reactors are provided in
parallel so that the second or next reactor is filled when the first or preceding reactor is
completing its last step.

20. SUMMARY
Depending on flow conditions and mixing of wastewater there in various
types of reactors are employed for wastewater treatment.
Various types of reactors and reactions occurring in reactors used for the
treatment of domestic wastewater.
In practice, a reactor with large length to small width ratio is assumed as
PFR.
When operating under similar conditions of flow and reaction order to
achieve the same degree of treatment, plug flow reactor requires less
volume than complete mix reactor.
Use of PBR, FBR and SBR for treating domestic wastewater is gaining
popularity.
Normally for biological processes, reactions are heterogeneous in nature
and of first order type.

21. T H A N K Y O
U